Instacart Market Basket Analysis

Introduction

Overview and Motivation

Our inspiration was from Kaggle competition - Instacart Market Basket Analysis which is also the data sets’ resource. Instacart is a grocery ordering and delivery application. They provide an anonymized dataset that contains a sample of over 3 million grocery orders from more than 200,000 Instacart’s users, and for each user, they provide between 4 and 100 of their orders, with the sequence of products purchased in each order, the week and hour of day the order was placed, and a relative measure of time between orders (details of each data set will be introduced below).

Instacart hopes campaign participants test models for predicting products that a user will buy again, try for the first time or add to cart next during a session, which may need to use the the models such as XGBoost, word2vec and Annoy (Jeremy Stanley, May 3, 2017).

Repurchase predicting and order placement day predicting are the popular and helpful predictions among e-commerce companies. For example, it can be applied to do personalization, managing supply and demand, churn prediction, improved customer service, etc (Bigcommerce Blog, Nick Shaw). Amazon has already developed a patent called “anticipatory shipping” that can predict what and when people want to buy and ship packages even before customers have placed an order (The Economic Time, Jan 27, 2014). In this case, they can largely optimizing logistics management, human and equipment resources and inventory arrangement, so that it would help them to decrease cost and increase profit. Meantime, this type of prediction also requires much more information of customers’ behavior, such as items customers have searched for, the amount of time a user’s cursor hovers over a product, times of clicks by users, purchase conversion rate of users’ click, add to cart, collection and so on.

In this case, since there are limitation of information and we would like to apply what models we have learnt in the course, we prefer to predict the day of the week that the order will be placed. Then, this would be an additional predictor to support the demand forecasting which is useful to make a right direction in the decision-making process, like inventory arrangement, for the e-commerce platform.

Research questions

Overall, we produce a new dataset based on what we have downloaded from the competition website, and assume that:

one order = one user (as the data limitation in the data set mentioned above);
we have already known what customers will buy in the next time, which means we have already known the demand.

Thus, our research questions will be:

What day of the week a given order will be placed?
For this question, we will use supervised methods.
Are there any common components between departments or aisles?
For this question, we will use unsupervised methods.

Exploratory Data Analysis

Data Description

Table 1 - aisles

The aisle table shows aisle unique ids under aisle_id column and aisle names under aisle column, for example aisle_id of 1 represents the prepared soup salads aisle. There are 134 ids in total and n/a is not found in the table.

The aisles table
aisle_id	aisle
1	prepared soups salads
2	specialty cheeses
3	energy granola bars
4	instant foods
5	marinades meat preparation
6	other
7	packaged meat
8	bakery desserts
9	pasta sauce
10	kitchen supplies
11	cold flu allergy
12	fresh pasta
13	prepared meals
14	tofu meat alternatives
15	packaged seafood
16	fresh herbs
17	baking ingredients
18	bulk dried fruits vegetables
19	oils vinegars
20	oral hygiene
21	packaged cheese
22	hair care
23	popcorn jerky
24	fresh fruits
25	soap
26	coffee
27	beers coolers
28	red wines
29	honeys syrups nectars
30	latino foods
31	refrigerated
32	packaged produce
33	kosher foods
34	frozen meat seafood
35	poultry counter
36	butter
37	ice cream ice
38	frozen meals
39	seafood counter
40	dog food care
41	cat food care
42	frozen vegan vegetarian
43	buns rolls
44	eye ear care
45	candy chocolate
46	mint gum
47	vitamins supplements
48	breakfast bars pastries
49	packaged poultry
50	fruit vegetable snacks
51	preserved dips spreads
52	frozen breakfast
53	cream
54	paper goods
55	shave needs
56	diapers wipes
57	granola
58	frozen breads doughs
59	canned meals beans
60	trash bags liners
61	cookies cakes
62	white wines
63	grains rice dried goods
64	energy sports drinks
65	protein meal replacements
66	asian foods
67	fresh dips tapenades
68	bulk grains rice dried goods
69	soup broth bouillon
70	digestion
71	refrigerated pudding desserts
72	condiments
73	facial care
74	dish detergents
75	laundry
76	indian foods
77	soft drinks
78	crackers
79	frozen pizza
80	deodorants
81	canned jarred vegetables
82	baby accessories
83	fresh vegetables
84	milk
85	food storage
86	eggs
87	more household
88	spreads
89	salad dressing toppings
90	cocoa drink mixes
91	soy lactosefree
92	baby food formula
93	breakfast bakery
94	tea
95	canned meat seafood
96	lunch meat
97	baking supplies decor
98	juice nectars
99	canned fruit applesauce
100	missing
101	air fresheners candles
102	baby bath body care
103	ice cream toppings
104	spices seasonings
105	doughs gelatins bake mixes
106	hot dogs bacon sausage
107	chips pretzels
108	other creams cheeses
109	skin care
110	pickled goods olives
111	plates bowls cups flatware
112	bread
113	frozen juice
114	cleaning products
115	water seltzer sparkling water
116	frozen produce
117	nuts seeds dried fruit
118	first aid
119	frozen dessert
120	yogurt
121	cereal
122	meat counter
123	packaged vegetables fruits
124	spirits
125	trail mix snack mix
126	feminine care
127	body lotions soap
128	tortillas flat bread
129	frozen appetizers sides
130	hot cereal pancake mixes
131	dry pasta
132	beauty
133	muscles joints pain relief
134	specialty wines champagnes

Table 2 - departments

The department table shows department unique ids under department_id column and department names under department_id column, for example department_id of 1 represents the frozen department. There are 21 ids in total and n/a is not found in the table.

The departments table
department_id	department
1	frozen
2	other
3	bakery
4	produce
5	alcohol
6	international
7	beverages
8	pets
9	dry goods pasta
10	bulk
11	personal care
12	meat seafood
13	pantry
14	breakfast
15	canned goods
16	dairy eggs
17	household
18	babies
19	snacks
20	deli
21	missing

Table 3 - products

The product table shows product unique ids under product_id column and department names under product_name column, for example product_id of 1 represents Chocolate Sandwich Cookies. This table also shows aisle_id and department_id that are associated with the product as well. There are approximately 50k ids in total and n/a is not found in the table.

The products table
product_id	product_name	aisle_id	department_id
1	Chocolate Sandwich Cookies	61	19
2	All-Seasons Salt	104	13
3	Robust Golden Unsweetened Oolong Tea	94	7
4	Smart Ones Classic Favorites Mini Rigatoni With Vodka Cream Sauce	38	1
5	Green Chile Anytime Sauce	5	13
6	Dry Nose Oil	11	11
7	Pure Coconut Water With Orange	98	7
8	Cut Russet Potatoes Steam N’ Mash	116	1
9	Light Strawberry Blueberry Yogurt	120	16
10	Sparkling Orange Juice & Prickly Pear Beverage	115	7
11	Peach Mango Juice	31	7
12	Chocolate Fudge Layer Cake	119	1
13	Saline Nasal Mist	11	11
14	Fresh Scent Dishwasher Cleaner	74	17
15	Overnight Diapers Size 6	56	18
16	Mint Chocolate Flavored Syrup	103	19
17	Rendered Duck Fat	35	12
18	Pizza for One Suprema Frozen Pizza	79	1
19	Gluten Free Quinoa Three Cheese & Mushroom Blend	63	9
20	Pomegranate Cranberry & Aloe Vera Enrich Drink	98	7
21	Small & Medium Dental Dog Treats	40	8
22	Fresh Breath Oral Rinse Mild Mint	20	11
23	Organic Turkey Burgers	49	12
24	Tri-Vi-SolÂ® Vitamins A-C-and D Supplement Drops for Infants	47	11
25	Salted Caramel Lean Protein & Fiber Bar	3	19
26	Fancy Feast Trout Feast Flaked Wet Cat Food	41	8
27	Complete Spring Water Foaming Antibacterial Hand Wash	127	11
28	Wheat Chex Cereal	121	14
29	Fresh Cut Golden Sweet No Salt Added Whole Kernel Corn	81	15
30	Three Cheese Ziti, Marinara with Meatballs	38	1
31	White Pearl Onions	123	4
32	Nacho Cheese White Bean Chips	107	19
33	Organic Spaghetti Style Pasta	131	9
34	Peanut Butter Cereal	121	14
35	Italian Herb Porcini Mushrooms Chicken Sausage	106	12
36	Traditional Lasagna with Meat Sauce Savory Italian Recipes	38	1
37	Noodle Soup Mix With Chicken Broth	69	15
38	Ultra Antibacterial Dish Liquid	100	21
39	Daily Tangerine Citrus Flavored Beverage	64	7
40	Beef Hot Links Beef Smoked Sausage With Chile Peppers	106	12
41	Organic Sourdough Einkorn Crackers Rosemary	78	19
42	Biotin 1000 mcg	47	11
43	Organic Clementines	123	4
44	Sparkling Raspberry Seltzer	115	7
45	European Cucumber	83	4
46	Raisin Cinnamon Bagels 5 count	58	1
47	Onion Flavor Organic Roasted Seaweed Snack	66	6
48	School Glue, Washable, No Run	87	17
49	Vegetarian Grain Meat Sausages Italian - 4 CT	14	20
50	Pumpkin Muffin Mix	105	13

Table 4 - order_products_train

This table shows the details of all order in the training data set provided by Instacart. It shows product_ids that are purchased in each order. For example, the order_id of 1 consists of 8 products, including the following product ids: 13176, 47209, 22035, etc. Also, there are ~131k orders in total and there is no n/a in the table.

The order_products_train table
order_id	product_id	add_to_cart_order	reordered
1	49302	1	1
1	11109	2	1
1	10246	3	0
1	49683	4	0
1	43633	5	1
1	13176	6	0
1	47209	7	0
1	22035	8	1
36	39612	1	0
36	19660	2	1
36	49235	3	0
36	43086	4	1
36	46620	5	1
36	34497	6	1
36	48679	7	1
36	46979	8	1
38	11913	1	0
38	18159	2	0
38	4461	3	0
38	21616	4	1
38	23622	5	0
38	32433	6	0
38	28842	7	0
38	42625	8	0
38	39693	9	0
96	20574	1	1
96	30391	2	0
96	40706	3	1
96	25610	4	0
96	27966	5	1
96	24489	6	1
96	39275	7	1
98	8859	1	1
98	19731	2	1
98	43654	3	1
98	13176	4	1
98	4357	5	1
98	37664	6	1
98	34065	7	1
98	35951	8	1
98	43560	9	1
98	9896	10	1
98	27509	11	1
98	15455	12	1
98	27966	13	1
98	47601	14	1
98	40396	15	1
98	35042	16	1
98	40986	17	1
98	1939	18	1

Table 5 - purchase time per order table

This table shows the purchase time (day of week under order_dow column and hour of day under order_hour_of_day column) for each order. For example, the order_id of 1187899 has order_dow of 4 and order_hour of day of 8. This means this order was maded on Thursday (order_dow = 4) at 8am (order_hour = 8).

The purchase time per order table
order_id	order_dow	order_hour_of_day
1187899	4	8
1492625	1	11
2196797	0	11
525192	2	11
880375	1	14
1094988	6	10
1822501	0	19
1827621	0	21
2316178	2	19
2180313	3	10
2461523	6	9
1854765	1	12
3402036	1	12
965160	0	16
2614670	5	14
3110252	4	11
62370	2	13
698604	4	13
1524161	0	13
3173750	0	9
2032076	0	20
2803975	0	11
1864787	5	11
2436259	0	12
1947848	4	20
2906490	4	22
2924697	5	18
519514	4	12
1750084	3	9
1647290	4	16
3088145	2	10
39325	2	18
13318	1	9
1651215	0	12
1019719	2	12
2989905	6	8
2639013	0	13
1072954	6	17
34647	3	19
2757217	0	11
669729	5	12
3038639	5	13
2608424	2	14
482516	4	7
3294399	4	8
1700658	6	11
21708	0	6
2178718	2	8
1734166	5	18
859654	1	10

Table 6 - user_purchases

This table joins Table 1 - 5 together. Thus, this table will include all necessary information that we need in the analysis, including order_id, purchase time, aisle and department. Please note that we do not include product_id and product_name in this table because the dimensionality is too large (over 50k categories). Thus, in our analysis, we will mainly use department_id (21 cetegories) in our analysis and use aisle_id (134 categories) in the PCA part in the unsupervised learning section.

The user purchases table
order_id	order_dow	order_hour_of_day	aisle_id	aisle	department_id	department
1187899	4	8	77	soft drinks	7	beverages
1187899	4	8	21	packaged cheese	16	dairy eggs
1187899	4	8	120	yogurt	16	dairy eggs
1187899	4	8	54	paper goods	17	household
1187899	4	8	45	candy chocolate	19	snacks
1187899	4	8	117	nuts seeds dried fruit	19	snacks
1187899	4	8	121	cereal	14	breakfast
1187899	4	8	23	popcorn jerky	19	snacks
1187899	4	8	84	milk	16	dairy eggs
1187899	4	8	53	cream	16	dairy eggs
1187899	4	8	77	soft drinks	7	beverages
1492625	1	11	96	lunch meat	20	deli
1492625	1	11	58	frozen breads doughs	1	frozen
1492625	1	11	107	chips pretzels	19	snacks
1492625	1	11	23	popcorn jerky	19	snacks
1492625	1	11	24	fresh fruits	4	produce
1492625	1	11	24	fresh fruits	4	produce
1492625	1	11	24	fresh fruits	4	produce
1492625	1	11	24	fresh fruits	4	produce
1492625	1	11	24	fresh fruits	4	produce
1492625	1	11	24	fresh fruits	4	produce
1492625	1	11	24	fresh fruits	4	produce
1492625	1	11	91	soy lactosefree	16	dairy eggs
1492625	1	11	46	mint gum	19	snacks
1492625	1	11	96	lunch meat	20	deli
1492625	1	11	80	deodorants	11	personal care
1492625	1	11	1	prepared soups salads	20	deli
1492625	1	11	38	frozen meals	1	frozen
1492625	1	11	38	frozen meals	1	frozen
1492625	1	11	38	frozen meals	1	frozen
1492625	1	11	38	frozen meals	1	frozen
1492625	1	11	38	frozen meals	1	frozen
1492625	1	11	38	frozen meals	1	frozen
1492625	1	11	38	frozen meals	1	frozen
1492625	1	11	69	soup broth bouillon	15	canned goods
1492625	1	11	37	ice cream ice	1	frozen
1492625	1	11	37	ice cream ice	1	frozen
1492625	1	11	37	ice cream ice	1	frozen
1492625	1	11	117	nuts seeds dried fruit	19	snacks
1492625	1	11	3	energy granola bars	19	snacks
1492625	1	11	69	soup broth bouillon	15	canned goods
1492625	1	11	69	soup broth bouillon	15	canned goods
2196797	0	11	29	honeys syrups nectars	13	pantry
2196797	0	11	24	fresh fruits	4	produce
2196797	0	11	21	packaged cheese	16	dairy eggs
2196797	0	11	66	asian foods	6	international
2196797	0	11	101	air fresheners candles	17	household
2196797	0	11	83	fresh vegetables	4	produce
2196797	0	11	66	asian foods	6	international
2196797	0	11	123	packaged vegetables fruits	4	produce

Visualization

Distribution of the order by time of purchase

The plot shows the distribution of the order by time (dow and hour).

The purchase time per order table
order_id	order_dow	order_hour_of_day
1187899	4	8
1492625	1	11
2196797	0	11
525192	2	11
880375	1	14
1094988	6	10
1822501	0	19
1827621	0	21
2316178	2	19
2180313	3	10
2461523	6	9
1854765	1	12
3402036	1	12
965160	0	16
2614670	5	14
3110252	4	11
62370	2	13
698604	4	13
1524161	0	13
3173750	0	9
2032076	0	20
2803975	0	11
1864787	5	11
2436259	0	12
1947848	4	20
2906490	4	22
2924697	5	18
519514	4	12
1750084	3	9
1647290	4	16
3088145	2	10
39325	2	18
13318	1	9
1651215	0	12
1019719	2	12
2989905	6	8
2639013	0	13
1072954	6	17
34647	3	19
2757217	0	11
669729	5	12
3038639	5	13
2608424	2	14
482516	4	7
3294399	4	8
1700658	6	11
21708	0	6
2178718	2	8
1734166	5	18
859654	1	10

We can observe on the left chart oder_dow that the most frequent days of ordering are Sunday’s and Monday’s comparing to the rest of the week, and on the right chart order_hour_of_day,we note a high demand of orders between 9am to 6pm.

Top 10 number of purchase by aisle

This table shows the top 10 aisles by the number of purchase. We can see that the most purchase aisles are fresh vegetables and fresh fruits (~150k orders each).

The top 10 number of purchase by aisle
aisle	department	total_order
fresh vegetables	produce	150609
fresh fruits	produce	150473
packaged vegetables fruits	produce	78493
yogurt	dairy eggs	55240
packaged cheese	dairy eggs	41699
water seltzer sparkling water	beverages	36617
milk	dairy eggs	32644
chips pretzels	snacks	31269
soy lactosefree	dairy eggs	26240
bread	bakery	23635

The number of purchase by department

This table shows the top 10 departments by the number of purchase. We can see that the most purchase aisles is produce (~409k orders).

The top 10 number of purchase by department
department	total_order
produce	409087
dairy eggs	217051
snacks	118862
beverages	114046
frozen	100426
pantry	81242
bakery	48394
canned goods	46799
deli	44291
dry goods pasta	38713

Sales Patterns

Here, we would like to observe the pattern of sales in depth by splitting into departments. First, it is the pattern of weekly sales.

From these graphs, we could observe the patterns as follow:

Most of the departments, except Alcohol, have similar pattern. The peak of the numbers of purchase are on Sunday and Monday, and tend to decrease during the weekday, and then start to increase on Friday.
For Alcohol, the figure increases slightly from the trough on Monday and reaches the top on Friday, then decreases sharply on Saturday.

PCA by department

We will use PCA to analyze whether if we could reduce dimension of the data set (The number of order by department).

PCA explains the similarity of variables. There are two metrics which are correlation(scaled) and covariance(non scaled). In our analysis, we focus on the relationship between the number of order from each department and day-of-week that users purchase. Thus, we will focus our PCA analysis on non-scale, i.e. using covariance. However, it would be interesting to see the differences of the results between scale and non-scaled PCAs as well, so we will also perform the PCA analysis with correlations.

Non-scaled PCA (Covariance)

We observe that the first and second components explain 46.7% and 13.8% of variance of the data. Referring to the rule of thumb which selects the number of dimensions that allow to explain at least 75% of the variation, therefore comp1 - comp5 are selected and around 79.8% of variance of the data are explained.

Our findings are as follows:

Produce has the highest variation. Also, it is highly positively correlated with Dim1 and negatively correlated with Dim2
The other departments including the second to sixth largest variance variables(Dairy egg, Snacks, Frozen, Beverages and Pantry) are positively correlated with Dim1 and Dim2.

Variance contribution from non-scaled PCA
	eigenvalue	percentage of variance	cumulative percentage of variance
comp 1	12.642	46.69	46.7
comp 2	3.727	13.76	60.5
comp 3	2.130	7.87	68.3
comp 4	1.577	5.82	74.1
comp 5	1.535	5.67	79.8
comp 6	1.115	4.12	83.9
comp 7	0.647	2.39	86.3
comp 8	0.610	2.25	88.6
comp 9	0.515	1.90	90.5
comp 10	0.469	1.73	92.2

Scaled PCA (Correlation)

We find that the first and second components can explain only 13.6% and 6.6% respectively, and we need 15 components (out of 21) to explain 75% of the variation. This means that correlations between departments are very low and we cannot use PCA to reduce the dimensions of the scaled data.

Variance contribution from non-scaled PCA
	eigenvalue	percentage of variance	cumulative percentage of variance
comp 1	2.861	13.62	13.6
comp 2	1.382	6.58	20.2
comp 3	1.167	5.55	25.8
comp 4	1.049	5.00	30.8
comp 5	1.035	4.93	35.7
comp 6	1.008	4.80	40.5
comp 7	0.990	4.71	45.2
comp 8	0.972	4.63	49.8
comp 9	0.944	4.49	54.3
comp 10	0.931	4.43	58.8
comp 11	0.903	4.30	63.1
comp 12	0.874	4.16	67.2
comp 13	0.871	4.15	71.4
comp 14	0.839	4.00	75.4
comp 15	0.807	3.84	79.2

Supervised Learning

Data Preparation for Models

Before starting applying the models to the data, we have decided to aggregate the column called id_orders by department, so we could know the number of products purchased by department. In addition, we have considered to keep the column order_dow, to identify on which day of the week an order was purchased.

After creating this new table, we converted the column order_dow from numeric(int) to categorical values(factor), and to understand better this values, we change the integer values to the name of the day of the week. For example: The value “0” was transformed to “Sunday”, “1” to “Monday”, “2” to “Tuesday”, and so on.

Moreover, we have decided to split our new table into two, to ensure that the model will not overfit the data and that the results of the predictions are good. To do so, we select for the first set; our training set, 80% of the observations randomly(around 105k obs), and for the observations that remain we took them as our test set(around 26k obs).

Number of products purchased by department per order
order_dow	canned.goods	dairy.eggs	produce	beverages	deli	frozen	pantry	snacks	bakery	household	meat.seafood	personal.care	dry.goods.pasta	babies	missing	other	breakfast	international	alcohol	bulk
Thursday	1	3	4	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
Saturday	0	3	3	1	1	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
Saturday	0	0	6	0	0	1	1	1	0	0	0	0	0	0	0	0	0	0	0	0
Saturday	0	0	4	0	1	2	0	0	0	0	0	0	0	0	0	0	0	0	0	0
Wednesday	8	11	7	4	3	3	4	1	1	5	1	1	0	0	0	0	0	0	0	0
Friday	0	0	4	1	0	0	0	3	0	0	0	2	1	0	0	0	0	0	0	0
Sunday	0	2	9	0	0	0	1	3	1	0	0	0	1	0	0	0	0	0	0	0
Sunday	0	0	4	1	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
Sunday	0	2	3	1	0	0	0	0	0	3	0	1	0	1	1	1	0	0	0	0
Wednesday	0	4	1	2	0	0	0	0	0	1	3	0	0	0	0	0	0	0	0	0
Saturday	0	2	4	0	0	0	0	1	0	0	0	0	0	0	0	0	1	0	0	0
Sunday	0	0	2	0	0	1	3	0	0	1	2	0	0	0	0	0	0	0	0	0
Saturday	0	2	1	0	0	1	0	0	0	0	0	0	3	1	0	0	0	0	0	0
Wednesday	0	4	7	0	1	0	2	0	0	0	0	0	0	0	1	0	0	2	0	0
Monday	0	0	0	0	0	0	0	0	0	1	0	0	0	0	0	0	0	0	0	0
Saturday	0	0	4	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
Sunday	0	0	0	0	0	1	1	0	0	0	0	0	0	0	0	0	0	1	0	0
Saturday	0	0	5	0	0	1	0	0	0	0	0	0	1	0	0	0	0	0	0	0
Tuesday	0	4	19	0	0	2	0	1	0	0	1	0	0	0	0	0	1	0	0	0
Thursday	0	0	0	1	0	0	0	0	0	0	0	0	1	0	0	0	0	0	0	0
Monday	1	1	0	5	1	1	0	3	0	2	0	0	0	0	0	0	0	0	0	0
Tuesday	0	1	0	0	0	1	0	0	0	1	0	0	0	0	0	1	0	0	0	0
Thursday	0	0	2	0	0	0	0	1	0	0	0	0	0	0	0	0	0	1	0	0
Monday	0	2	7	10	1	1	1	0	0	0	0	0	2	0	0	0	0	1	0	0
Monday	1	2	2	0	0	0	0	2	0	0	1	0	0	0	0	0	0	0	0	0
Saturday	0	0	3	1	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
Wednesday	0	0	3	0	0	2	0	0	0	0	0	0	0	0	0	0	0	0	0	0
Monday	0	1	4	0	0	0	0	0	0	0	1	0	0	0	0	0	0	0	0	0
Sunday	0	0	0	0	0	0	0	2	0	0	0	0	0	0	0	0	0	0	0	0
Friday	0	0	3	0	2	0	3	0	1	0	0	0	0	0	0	0	0	0	0	0
Sunday	0	0	9	3	0	0	0	0	1	0	0	2	2	0	0	0	0	0	0	0
Friday	1	5	3	0	0	7	0	4	3	0	1	0	3	0	0	0	1	2	0	0
Sunday	0	0	2	0	0	0	2	0	0	0	1	0	0	0	0	0	0	0	0	0
Thursday	0	5	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
Friday	0	1	0	1	0	0	1	1	0	0	0	0	0	0	0	0	0	0	0	0
Wednesday	0	1	1	2	0	0	0	0	0	0	0	0	0	0	0	0	0	0	1	0
Monday	0	3	6	0	0	0	0	0	0	0	0	0	0	0	0	0	1	0	0	0
Wednesday	0	0	5	0	0	1	0	0	0	1	1	0	0	0	0	0	0	0	0	0
Thursday	0	0	0	5	0	0	1	3	0	1	0	0	0	0	0	1	0	0	0	0
Sunday	0	1	14	0	1	0	1	0	0	0	0	0	0	0	0	0	0	0	0	0
Tuesday	0	0	2	2	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
Monday	0	2	7	3	1	1	2	0	1	0	0	1	0	0	0	0	1	0	0	0
Friday	0	0	0	3	0	0	0	0	0	0	1	0	0	0	1	0	0	0	0	0
Tuesday	0	4	6	3	2	3	1	7	1	0	1	0	0	0	1	0	2	0	0	0
Monday	0	4	3	1	0	0	1	1	0	0	0	0	0	0	0	0	0	0	0	0
Friday	3	1	7	1	0	0	6	0	0	0	0	1	1	0	0	0	0	0	0	0
Monday	0	0	3	0	0	0	0	0	0	0	0	0	1	0	0	0	0	0	0	0
Tuesday	2	0	4	0	0	0	1	0	0	1	0	0	0	0	0	0	0	0	0	0
Tuesday	0	2	1	0	0	1	0	0	0	1	0	0	1	0	0	0	0	0	0	0
Friday	0	3	0	0	0	0	0	1	0	0	0	0	1	0	0	0	0	0	0	1

Models

Our goal is to determine which day of the week a given order will be placed. Since we have transformed the column order_dow as a factor with categorical values, we will apply models that consider a classification task.

We have chosen the models as follows:

Decision Trees
Random Forest
Multinomial Logistic Regression
Logistic Regression

In addition, we will implement to each of the models some of the following approaches:

One day of the week - Unbalanced data
One day of the week - Balanced data with Sub-sampling and Cross-Validation
Weekdays and Weekend - Balanced data with Sub-sampling and Cross-Validation

Decision Trees - Classification

Decision trees are algorithms that recursively search the space for the best boundary possible, until we unable them to do so (Ivo Bernardo,2021). The basic functionality of decision trees is to split the data space into rectangles, by measuring each split. The main goal is to minimize the impurity of each split from the previous one.

One day of the week - Unbalanced data

For this approach we want to measure the accuracy of the model with the unbalanced data. Furthermore, it will be interesting to see which departments were considered the best to split the data into days of the week to later be compared to a balanced data with cross-validation (second approach).

According to the pruned tree, we observe that the department produce have the most relevance within the departments, this could be influenced by the fact that this department has the highest number of products purchased in our data set. Furthermore, the tree show us that with an amount of products purchased higher or equal than 3, the model will classify the day of the week as Sunday, if it is lower than 3 the tree will split into another node containing the frozen department.

Likewise, the same procedure will be consider for this node and the following, they will start from the previous node and will try to minimize the impurity at each split. It should be noted that we cannot observed on the terminal nodes all the days of the week, because of the way in which the trees are generated. For the same reason, we expect on the prediction of the test set, a prediction value of “0” on the days of the week different to Sunday and Monday.

#> Confusion Matrix and Statistics
#> 
#>            Reference
#> Prediction  Friday Monday Saturday Sunday Thursday Tuesday Wednesday
#>   Friday         0      0        0      0        0       0         0
#>   Monday       694    706      578    650      708     685       661
#>   Saturday       0      0        0      0        0       0         0
#>   Sunday      2787   3228     3202   4843     2483    2538      2476
#>   Thursday       0      0        0      0        0       0         0
#>   Tuesday        0      0        0      0        0       0         0
#>   Wednesday      0      0        0      0        0       0         0
#> 
#> Overall Statistics
#>                                         
#>                Accuracy : 0.211         
#>                  95% CI : (0.207, 0.216)
#>     No Information Rate : 0.209         
#>     P-Value [Acc > NIR] : 0.2           
#>                                         
#>                   Kappa : 0.016         
#>                                         
#>  Mcnemar's Test P-Value : NA            
#> 
#> Statistics by Class:
#> 
#>                      Class: Friday Class: Monday Class: Saturday
#> Sensitivity                  0.000        0.1795           0.000
#> Specificity                  1.000        0.8217           1.000
#> Pos Pred Value                 NaN        0.1508             NaN
#> Neg Pred Value               0.867        0.8503           0.856
#> Prevalence                   0.133        0.1499           0.144
#> Detection Rate               0.000        0.0269           0.000
#> Detection Prevalence         0.000        0.1784           0.000
#> Balanced Accuracy            0.500        0.5006           0.500
#>                      Class: Sunday Class: Thursday Class: Tuesday
#> Sensitivity                  0.882           0.000          0.000
#> Specificity                  0.194           1.000          1.000
#> Pos Pred Value               0.225             NaN            NaN
#> Neg Pred Value               0.861           0.878          0.877
#> Prevalence                   0.209           0.122          0.123
#> Detection Rate               0.185           0.000          0.000
#> Detection Prevalence         0.822           0.000          0.000
#> Balanced Accuracy            0.538           0.500          0.500
#>                      Class: Wednesday
#> Sensitivity                      0.00
#> Specificity                      1.00
#> Pos Pred Value                    NaN
#> Neg Pred Value                   0.88
#> Prevalence                       0.12
#> Detection Rate                   0.00
#> Detection Prevalence             0.00
#> Balanced Accuracy                0.50

As expected, only on Sunday and Monday we get predictive values for all the days of the week, while in the rest we get zero. Overall, the accuracy of this model is low with a score of 0.21, meaning that the model have ((0.21 - (1/7))= 0.077) around 8% of accuracy classifying the days of the week. It is important to recall that there is a big difference between sensitivity and specificity because our data is not balanced.

One day of the week - Balanced data with Sub-sampling and Cross-Validation

Now for this approach, we will balanced the data with sub-sampling and make the overall score more robust by applying to the model a cross-validation technique, this will help us to find the best set of hyperparameters.

#>  .outcome  Fri Mon Sat Sun Thu Tue Wed                                    cover
#>  Saturday [.14 .13 .16 .14 .14 .14 .15] when produce <  3 & frozen >= 1     18%
#>    Sunday [.13 .15 .15 .18 .13 .13 .13] when produce >= 3                   44%
#>  Thursday [.15 .14 .13 .11 .16 .15 .16] when produce <  3 & frozen <  1     38%

The left column (.outcome) of the rules show the day that was selected for the terminal node (the one with the highest probability) and next to it the probability of each day of the week for the department selected. In this case for the last rule it seems that Wednesday and Thursday have the same probability because of the rounding, but Thursday its 0.003 above Wednesday, this can be seen from the tree plot.

The rightmost column (cover) gives the percentage of observations in each rule. The first rule says that Saturday will be chosen when the department produce is lower than 3 and higher or equal than 1 with a probability of 18%. Then we can look at the results of the model in the Confusion Matrix.

#> Confusion Matrix and Statistics
#> 
#>            Reference
#> Prediction  Friday Monday Saturday Sunday Thursday Tuesday Wednesday
#>   Friday         0      0        0      0        0       0         0
#>   Monday         0      0        0      0        0       0         0
#>   Saturday     678    645      751    904      595     587       583
#>   Sunday      1454   1797     1768   2992     1235    1291      1259
#>   Thursday    1349   1492     1261   1597     1361    1345      1295
#>   Tuesday        0      0        0      0        0       0         0
#>   Wednesday      0      0        0      0        0       0         0
#> 
#> Overall Statistics
#>                                        
#>                Accuracy : 0.195        
#>                  95% CI : (0.19, 0.199)
#>     No Information Rate : 0.209        
#>     P-Value [Acc > NIR] : 1            
#>                                        
#>                   Kappa : 0.035        
#>                                        
#>  Mcnemar's Test P-Value : NA           
#> 
#> Statistics by Class:
#> 
#>                      Class: Friday Class: Monday Class: Saturday
#> Sensitivity                  0.000          0.00          0.1987
#> Specificity                  1.000          1.00          0.8223
#> Pos Pred Value                 NaN           NaN          0.1583
#> Neg Pred Value               0.867          0.85          0.8591
#> Prevalence                   0.133          0.15          0.1441
#> Detection Rate               0.000          0.00          0.0286
#> Detection Prevalence         0.000          0.00          0.1808
#> Balanced Accuracy            0.500          0.50          0.5105
#>                      Class: Sunday Class: Thursday Class: Tuesday
#> Sensitivity                  0.545          0.4265          0.000
#> Specificity                  0.576          0.6382          1.000
#> Pos Pred Value               0.254          0.1403            NaN
#> Neg Pred Value               0.827          0.8894          0.877
#> Prevalence                   0.209          0.1216          0.123
#> Detection Rate               0.114          0.0519          0.000
#> Detection Prevalence         0.450          0.3697          0.000
#> Balanced Accuracy            0.560          0.5324          0.500
#>                      Class: Wednesday
#> Sensitivity                      0.00
#> Specificity                      1.00
#> Pos Pred Value                    NaN
#> Neg Pred Value                   0.88
#> Prevalence                       0.12
#> Detection Rate                   0.00
#> Detection Prevalence             0.00
#> Balanced Accuracy                0.50

From the confusion matrix we observe a better result between the sensitivity and specificity across the classes, if we compare the previous model with this one, we notice that on the class Sunday the values of the sensitivity changed from 0.882 to 0.545, and for the specificity from 0.194 to 0.576. As expected, the Accuracy has decreased from 0.211 to 0.195, meaning that the model have ((0.195 - (1/7))= 0.052) around 5% of accuracy per day of the week, but the Balanced Accuracy is better. This model would be preferred than the one used with unbalanced data.

Weekdays and Weekend - Balanced data with Sub-sampling and Cross-Validation

For the Final approach we transformed the levels of the column order_dow into two, one for the days during the week and the remaining for the weekend. On top of that we balanced our levels “weekday” and “weekend” and consider a Cross-Validation to train the model.

We try to plot the final tree computed by the model, but it was not possible to interpret it, due to the overlapping nodes shown in the graph, but we could see that the departments produce, frozen and meat.seafood, were among the first splits.

The results of the confusion matrix show us a well balanced data from what we can observed in the sensitivity and specificity. The Accuracy of the model is similar comparing it with the other two approaches, the model have ((0.533 - (1/2))= 0.033) 3% of accuracy. Overall, all different approaches have a low score at predicting the day of the week based on the department purchases from previous orders.

Random Forest

Random Forest (RF) are algorithms of a set of decision trees that will produce a final prediction with the average outcome of the set of trees considered (user can define the amount of trees and the number of variables for each node). One of the reasons that we decided to test this method is because RF are considered to be more stable than Decision Trees; more trees better performance, but certain advantages come at a price. RF slow down the computation speed and cannot be visualize, however, we will look at the results for later comparison (Saikumar Talari, 2022).

Weekdays and Weekend - Balanced data with Sub-sampling and Cross-Validation

For this method we will consider the same approach as the last one of Classification Tree. We faced some computation speed problems while running the model, for that reason we decided to considered only 10,000 orders to reduce the waiting time.

As expected, the Accuracy of the model is higher than the Classification Tree as well as Cohen’s Kappa and the balanced accuracy. This model would be preferred at predicting “weekday” and “weekend” as it has better results.

Multinomial logistic regression

One day of the week - Unbalanced data

Multinomial logistic regression is a classification method that generalizes logistic regression to multiclass problems, i.e. with more than two possible discrete outcomes (Wikipedia,2021). Like binary logistic regression, multinomial logistic regression uses maximum likelihood estimation to evaluate the probability of categorical membership.

Our first approach is to predict the day of the week that the order will be placed according to the product composition in the order. Since there are 7 days in a week so it is not a binary logistic regression problem but a multinomial logistic regression problem.

We select Sunday as the reference level. To build the model, we use the number of products in each department of the order as explanatory variables.

#> Confusion Matrix and Statistics
#> 
#>            Reference
#> Prediction  Friday Monday Saturday Sunday Thursday Tuesday Wednesday
#>   Friday       129     99       91     79       93      94       102
#>   Monday        69     87       39     54       95      89        85
#>   Saturday      39     46       50     51       36      29        24
#>   Sunday      3208   3673     3584   5278     2931    2981      2900
#>   Thursday      20     19        9     21       24      18        18
#>   Tuesday        0      0        0      0        0       0         0
#>   Wednesday     16     10        7     10       12      12         8
#> 
#> Overall Statistics
#>                                         
#>                Accuracy : 0.213         
#>                  95% CI : (0.208, 0.218)
#>     No Information Rate : 0.209         
#>     P-Value [Acc > NIR] : 0.105         
#>                                         
#>                   Kappa : 0.01          
#>                                         
#>  Mcnemar's Test P-Value : <2e-16        
#> 
#> Statistics by Class:
#> 
#>                      Class: Friday Class: Monday Class: Saturday
#> Sensitivity                0.03706       0.02211         0.01323
#> Specificity                0.97548       0.98068         0.98998
#> Pos Pred Value             0.18777       0.16795         0.18182
#> Neg Pred Value             0.86882       0.85043         0.85634
#> Prevalence                 0.13267       0.14993         0.14406
#> Detection Rate             0.00492       0.00332         0.00191
#> Detection Prevalence       0.02618       0.01974         0.01048
#> Balanced Accuracy          0.50627       0.50140         0.50160
#>                      Class: Sunday Class: Thursday Class: Tuesday
#> Sensitivity                 0.9609        0.007521          0.000
#> Specificity                 0.0708        0.995444          1.000
#> Pos Pred Value              0.2149        0.186047            NaN
#> Neg Pred Value              0.8723        0.878705          0.877
#> Prevalence                  0.2093        0.121613          0.123
#> Detection Rate              0.2012        0.000915          0.000
#> Detection Prevalence        0.9358        0.004916          0.000
#> Balanced Accuracy           0.5158        0.501483          0.500
#>                      Class: Wednesday
#> Sensitivity                  0.002550
#> Specificity                  0.997100
#> Pos Pred Value               0.106667
#> Neg Pred Value               0.880408
#> Prevalence                   0.119555
#> Detection Rate               0.000305
#> Detection Prevalence         0.002858
#> Balanced Accuracy            0.499825

According to the confusion matrix, the accuracy(0.213) is low and there is a big difference between sensitivity and specificity in each class. For example, the sensitivity of class Friday is 0.037 while the specificity of class Friday is 0.975. Also the kappa(0.01) is very small which means the observed accuracy is only a little higher than the accuracy that one would expect from a random model. We try to balance the data and use a cross-validation to improve the model accuracy.

One day of the week - balanced data with cross-validation

Before balancing the data, We need to check the frequency of each class. The class Wednesday has the smallest frequency(12550). We will balance data by sub-sampling according to the frequency of class Wednesday.

#> 
#>    Sunday    Friday    Monday  Saturday  Thursday   Tuesday 
#>     21972     13925     15738     15121     12768     12896 
#> Wednesday 
#>     12550

#> 
#>    Sunday    Friday    Monday  Saturday  Thursday   Tuesday 
#>     12550     12550     12550     12550     12550     12550 
#> Wednesday 
#>     12550

We only sub-sample the data without applying cross-validation. Now every class has the same frequency(12550).

We try the cross-validation with the sub-sampling data by using the train function of caret package, but the data set is too big and it takes a very long time to run it so we decide not to include the cross-validation.

#> Confusion Matrix and Statistics
#> 
#>            Reference
#> Prediction  Friday Monday Saturday Sunday Thursday Tuesday Wednesday
#>   Friday       245    258      222    297      203     203       200
#>   Monday       348    448      383    629      310     347       329
#>   Saturday     403    436      459    684      348     345       336
#>   Sunday       797   1033     1064   1868      664     735       698
#>   Thursday     737    760      715    869      718     709       689
#>   Tuesday      356    371      360    453      320     282       290
#>   Wednesday    595    628      577    693      628     602       595
#> 
#> Overall Statistics
#>                                         
#>                Accuracy : 0.176         
#>                  95% CI : (0.171, 0.181)
#>     No Information Rate : 0.209         
#>     P-Value [Acc > NIR] : 1             
#>                                         
#>                   Kappa : 0.03          
#>                                         
#>  Mcnemar's Test P-Value : <2e-16        
#> 
#> Statistics by Class:
#> 
#>                      Class: Friday Class: Monday Class: Saturday
#> Sensitivity                0.07038        0.1139          0.1214
#> Specificity                0.93923        0.8948          0.8864
#> Pos Pred Value             0.15049        0.1603          0.1524
#> Neg Pred Value             0.86851        0.8513          0.8570
#> Prevalence                 0.13267        0.1499          0.1441
#> Detection Rate             0.00934        0.0171          0.0175
#> Detection Prevalence       0.06205        0.1065          0.1148
#> Balanced Accuracy          0.50481        0.5044          0.5039
#>                      Class: Sunday Class: Thursday Class: Tuesday
#> Sensitivity                 0.3401          0.2250         0.0875
#> Specificity                 0.7594          0.8057         0.9066
#> Pos Pred Value              0.2723          0.1382         0.1160
#> Neg Pred Value              0.8130          0.8825         0.8765
#> Prevalence                  0.2093          0.1216         0.1228
#> Detection Rate              0.0712          0.0274         0.0107
#> Detection Prevalence        0.2614          0.1981         0.0927
#> Balanced Accuracy           0.5497          0.5153         0.4970
#>                      Class: Wednesday
#> Sensitivity                    0.1897
#> Specificity                    0.8388
#> Pos Pred Value                 0.1378
#> Neg Pred Value                 0.8840
#> Prevalence                     0.1196
#> Detection Rate                 0.0227
#> Detection Prevalence           0.1646
#> Balanced Accuracy              0.5143

From the confusion matrix report we can notice that there is an improvement on the difference between sensitivity and specificity of each class. For example, the sensitivity and specificity of class Thursday are 0.007 and 0.995 in the previous model. After balancing the data, now the sensitivity and specificity of class Thursday are 0.231 and 0.801. The kappa is also higher(from 0.01 to 0.03)

Logistic regression

Weekdays and Weekend - Balanced data and Cross-Validation

The logistic regression is a regression adapted to binary classification. The basic idea of logistic regression is to use the mechanism already developed for linear regression by modeling the probability pi using a linear predictor function, i.e. a linear combination of the explanatory variables and a set of regression coefficients that are specific to the model at hand but the same for all trials. The linear combination is transformed to a probability using a sigmoid function.

In order to further improve our model quality, we think about aggregating the classes of the day of week. Usually the buying behavior is different between weekday and weekend. So we separate the day of week into two classes weekday and weekend.

Now the outcome variable has only two categories so we can use the binomial logistic regression.

According to the confusion matrix the balanced accuracy is higher and the difference between sensitivity(0.62) and specificity(0.48) is even smaller. Now the Kappa is 0.10, higher than Cohen’s Kappa previous model(0.03), and the Accuracy is 0.57.

Comparing the result of this model against the previous ones, we note that Random forest model is the one that resembles these results, only Logistic regression is slightly higher on the Accuracy by 0.008, on Cohen’sKappa by 0.010, and on the Balanced Accuracy by 0.005. So, for those reasons we have decided choosing this model over the rest.

Variable Importance

Variable importance is a method that provides a measure of the importance of each feature for the model prediction quality. We analyze the variables importance of our 4 models. There are 21 explanatory variables in each of our models and we only show the top 10 most important variables in the plots.

Note: We faced some computation speed problems while running the the Variable Importance for all the models, for that reason we decided to considered the same amount of observations as the Random Forest model(10k) to reduce the waiting time.

As we note from this chart the models Classification Tree, Random Forest, and Logistic Regression use the AUC loss to compare the model quality of shuffling different variables. AUC is a synthetic measure of the distance to random model in the ROC curve plot. The larger AUC, the better the model.

According to the feature importance of this three models, the most important department is the produce department. As we have seen above in the table of number of purchases per department, the produce department is the leading one with almost twice the number of the second one in that table. This could be one of the main reasons why all three models have chosen it as the most relevant. It is important to mention that if we shuffle this variable, the AUC of the model will have the largest loss.

For the model Multinomial Logistic Regression, we use the Root Mean Square Error(RMSE) to compare the model quality of shuffling different variables. According to the plot, the most important variable is dairy eggs. If we shuffle this variable, the RMSE of the model will have the largest increase.

Unsupervised learning

In this section, we will work with unsupervised learning methods i.e. Clustering and PCA to learn how we can reduce the dimensionality of the original data set and how to group data by similarity/or dissimilarity of all the features. Then, we will also study further by using a hybrid supervised/unsupervised learning method to perform the prediction (supervised) by using the result from the PCA analysis (unsupervised).

Clustering

In this section, we will study clustering approaches, Hierarchical clustering and Partitioning methods, to find groups of instances/observations that have similar features.

Due to the limitation of the clustering functions in R, the execution time when we tried to cluster all instances (131,206 instances) was very long. Since, in this exercise, we would like to focus on the approaches/methodologies, we will randomly choose only 1% of the instances (1,312 instances) to perform the analysis in order to reduce the execution time.

Hierarchical clustering

Distance

First, we apply an Agglomerative nesting (AGNES) and compute the distances using Euclidean distances because all our features are numerical.

Example of the Euclidean distance between instances
Var1	Var2	value
1	1	0.00
2	1	9.95
3	1	6.00
4	1	9.16
5	1	7.14
6	1	10.95
7	1	7.62
8	1	8.72
9	1	9.90
10	1	6.00
11	1	4.47
12	1	11.53
13	1	7.75
14	1	7.00
15	1	11.49
16	1	11.66
17	1	2.45
18	1	9.27
19	1	11.87
20	1	7.07
21	1	9.80
22	1	11.96
23	1	9.70
24	1	10.58
25	1	9.33
26	1	6.00
27	1	8.00
28	1	9.22
29	1	9.75
30	1	10.77
31	1	11.40
32	1	5.75
33	1	10.34
34	1	5.83
35	1	9.54
36	1	10.91
37	1	9.59
38	1	9.95
39	1	8.94
40	1	6.33
41	1	11.09
42	1	8.89
43	1	11.87
44	1	9.11
45	1	12.49
46	1	11.79
47	1	11.00
48	1	9.64
49	1	15.46
50	1	7.68

> Dendrogram

We then apply the dendrogram using the complete linkage to visualized the output of the hierarchical clustering. Since it’s difficult to read the original dendrogram, we will selected the optimal number of clusters and cut the tree branches in the next steps.

Choice of the number of clusters

We will choose the optimal number of clusters from statistics. We will apply the within-cluster sum of squares, the GAP statistics and the silhouette using complete linkage on Euclidean distance.

From the graph we can interpret the following:

The Within-Cluster Sum of Squares: It seems there is an elbow at the cluster number 3, for that reason we choose the optimal k = 3.
The Silhouette: With the silhouette statistic, the larger value of the average silhouette width is the better. We can see that the optimal number of clustering is 3.
The Gap Statistic: We see that the Gap statistic returns 2 clusters as an optimal number of cluster.

For those reasons we have decided to choose 3 as a number of clusters and cut the trees as follows.

Interpretation of the clusters

We will analyze the clusters by using the box plot for each feature.

Our observations are as follows:

Cluster 2 has very small canned goods, deli, meat.seafood and dry.goods.pasta, while Cluster 1 and Cluster 3 have relatively high amounts for these departments.
Cluster 3 has higher snacks, beverages, daily eggs and breakfast than Cluster 1.
Cluster 1 has the highest produce.

Partitioning methods

In this section, we will apply partitioning methods, K-means and Partitioning Around the Medoid (PAM). For the partitioning methods, we first need to identify the number of clusters and then use the chosen number of clusters to perform the analysis.

K-means

We will use WSS, silhouette and the Gap statistic to determine the number of clusters used for K-means. It’s important to note that K-means is suitable for numerical features only. Since all our features are numerical, it’s appropriate to perform the K-means analysis.

The Within-Cluster Sum of Squares: It seems there is an elbow at the cluster number 2, for that reason we choose the optimal k = 2.
Silhouette: From this approach, k = 2 (the highest number) is the optimal number of clusters.
Gap Statistic: From this graph, it’s not conclusive. The function chose 13 as an optimal number but 15 and 22, which are local maximum, might be also used here.

Therefore, the number 2 is an optimal number of clusters. Afterward, we plot the box plot to distinguish the characteristic of those 2 clusters. We observe that cluster 2 has higher average numbers of purchases than cluster 1 in every department that the median of numbers of purchases are higher than zero such as canned.good, dairy.eggs, produce, beverages, deli, frozen, pantry, snacks, bakery, meat.seafood and dry.goods.pasta.

Next, we will show a scatter plot along the first and second principal components and group by 2 clusters using K-means. We will see that PC1 can distinguish the clusters quite well. Cluster 1 has higher PC1 than Cluster 2.

Regarding the principal components, the percentages of variance of PC1 and PC2 are 47% and 14% respectively, which are close to what we observed from PCA analysis in the EDA section however, please note that the numbers are not the same because we chose only 1% of the features from the original data set for this clustering exercise due to the computer capacity limitation.

Partitioning Around the Medoid (PAM)

Similar to K-means, we will need to find an optimal number of clusters before performing the analysis. We will Silhouette to determine the optimal k.

From the graph below, we find that the optimal number of clusters is 2.

Then, we will plot silhouette to show the silhouettes of all the instances and the average silhouette.

From the graph, we see that cluster 2 is well formed (well separated from Cluster 1, with the average silhouette of 0.46). Cluster 1 is less homogeneous with an average silhouette of 0.16 only. The average silhouette of the data set is 0.4.

Afterward, we plot the box plot to distinguish the characteristic of those 2 clusters. We observe that cluster 1 has higher average numbers of purchases than cluster 2. It’s interesting to see that the characteristics between 2 clusters from PAM are very similar to the clusters from K-means.

PCA and hybrid supervised and unsupervised learning approach

In our data set, there are 134 aisles which are grouped into 21 departments. So far in our supervised learning approach, we focus on the number of purchase per department. In this section, we would like to combine supervised and unsupervised learning approaches as follows.

Grouping aisles by using PCA: Our assumption is that grouping aisles by PCA might better reflect a purchase pattern of customer than when grouping by department.
Performing supervised learning approach from the first step

PCA by aisle

Non-scaled PCA (Covariance)

We observe that the first and second components explain 23.98% and 9.02% of variance of the data. Referring to the rule of thumb which selects the number of dimensions that allow to explain at least 75% of the variation, therefore comp1 - comp28 are selected and around 75.5% of variance of the data are explained.

Variance contribution from non-scaled PCA
	eigenvalue	percentage of variance	cumulative percentage of variance
comp 1	4.211	23.985	24.0
comp 2	1.585	9.027	33.0
comp 3	0.889	5.061	38.1
comp 4	0.668	3.806	41.9
comp 5	0.531	3.022	44.9
comp 6	0.487	2.772	47.7
comp 7	0.442	2.516	50.2
comp 8	0.370	2.106	52.3
comp 9	0.350	1.993	54.3
comp 10	0.303	1.724	56.0
comp 11	0.293	1.671	57.7
comp 12	0.276	1.573	59.3
comp 13	0.266	1.513	60.8
comp 14	0.241	1.372	62.1
comp 15	0.217	1.233	63.4
comp 16	0.206	1.172	64.5
comp 17	0.194	1.102	65.6
comp 18	0.183	1.040	66.7
comp 19	0.181	1.029	67.7
comp 20	0.175	0.998	68.7
comp 21	0.170	0.967	69.7
comp 22	0.163	0.931	70.6
comp 23	0.152	0.867	71.5
comp 24	0.149	0.849	72.3
comp 25	0.144	0.818	73.1
comp 26	0.141	0.806	74.0
comp 27	0.139	0.790	74.7
comp 28	0.130	0.742	75.5

> Scaled PCA (Correlation)

We find that the first and second components can explain only 3.2% and 1.8% respectively, and we need 93 components (out of 134) to explain 75% of the variation. This means that correlations between aisles are very low and we cannot use PCA to reduce the dimensions of the scaled data.

Variance contribution from scaled PCA
	eigenvalue	percentage of variance	cumulative percentage of variance
comp 1	4.27	3.186	3.19
comp 2	2.43	1.814	5.00
comp 3	1.93	1.439	6.44
comp 4	1.68	1.257	7.70
comp 5	1.56	1.168	8.86
comp 6	1.46	1.088	9.95
comp 7	1.42	1.059	11.01
comp 8	1.33	0.990	12.00
comp 9	1.27	0.948	12.95
comp 10	1.25	0.937	13.88

All in all, we can see that scaled PCA cannot reduce dimensions of the data set. Although the disadvantage of the non-scaled PCA is that it tends to include information from variables that have high variance and high correlation to the others and the model tends to neglect variables that have low variance and low correlation, we can derive the benefit of PCA, which is to reduce the dimension. Also, our objective in our research is to support demand forecasting such as inventory arrangement. The high variation departments and aisles tend to more important to this purpose. Thus, we focus on non-scaled PCA.

A hybrid supervised and unsupervised learning approach

In this section, we will apply a supervised learning approach with the output from the PCA (non-scale) analysis. From the PCA analysis, we select the first 28 principal components, which can explain over 75% of the total variation. For the supervised learning approach that will be performed in this section, we will use the logistic regression which is the best model from the supervised learning section.

#> Confusion Matrix and Statistics
#> 
#>           Reference
#> Prediction weekday weekend
#>    weekday   10655    4978
#>    weekend    6313    4295
#>                                         
#>                Accuracy : 0.57          
#>                  95% CI : (0.564, 0.576)
#>     No Information Rate : 0.647         
#>     P-Value [Acc > NIR] : 1             
#>                                         
#>                   Kappa : 0.088         
#>                                         
#>  Mcnemar's Test P-Value : <2e-16        
#>                                         
#>             Sensitivity : 0.628         
#>             Specificity : 0.463         
#>          Pos Pred Value : 0.682         
#>          Neg Pred Value : 0.405         
#>              Prevalence : 0.647         
#>          Detection Rate : 0.406         
#>    Detection Prevalence : 0.596         
#>       Balanced Accuracy : 0.546         
#>                                         
#>        'Positive' Class : weekday       
#>

According to the result from the original data set (without PCA) in the supervised learning section, sensitivity, specificity, kappa and accuracy are 0.628, 0.479, 0.103 and 0.57 respectively.

From the confusion matrix of the hybrid approach, we find that the sensitivity (0.628) and the accuracy (0.57) are equivalent to the result from logistic regression. However, the specificity (0.463) and kappa (0.088) are slightly lower than the result of logistic regression. Thus, we conclude that this method doesn’t improve the quality of the model.

Conclusion

For the supervised learning model, we use sensitivity, specificity, balanced accuracy, overall accuracy and kappa to compare the quality of different models.

For the unsupervised learning model, we perform clustering and PCA analysis to group a set of instances/features that share some common characteristics. In addition, we also use the result from the PCA analysis to perform a hybrid supervised/unsupervised learning model to see if we can use PCA to improve the performance of the supervised learning model.

Supervised learning

One day of the week - Unbalanced data

According to table1, table2, the overall accuracy and kappa of decision tree and multinomial regression are very low. What’s more, there is a serious unbalance between sensitivity and specificity. The quality of both models is pretty similar and needs to be improved.

Accuracy: decision tree:O.211 multinomial logistic regression:O.213

kappa:decision tree:O.016 multinomial logistic regression:O.01

Table 1 - Decision Tree

decision tree

	Accuracy	Kappa
overall	0.211	0.016

	Sensitivity	Specificity	Balanced Accuracy
Class: Friday	0.000	1.000	0.500
Class: Monday	0.179	0.822	0.501
Class: Saturday	0.000	1.000	0.500
Class: Sunday	0.882	0.194	0.538
Class: Thursday	0.000	1.000	0.500
Class: Tuesday	0.000	1.000	0.500
Class: Wednesday	0.000	1.000	0.500

Table 2 - Multinomial Logistic Regression

multinomial logistic regression

	Accuracy	Kappa
overall	0.213	0.01

	Sensitivity	Specificity	Balanced Accuracy
Class: Friday	0.037	0.975	0.506
Class: Monday	0.022	0.981	0.501
Class: Saturday	0.013	0.990	0.502
Class: Sunday	0.961	0.071	0.516
Class: Thursday	0.008	0.995	0.501
Class: Tuesday	0.000	1.000	0.500
Class: Wednesday	0.003	0.997	0.500

One day of the week - balanced data

According to table3 and table4, after we balanced the data and do the cross-validation, the difference between sensitivity and specificity is smaller. Even though the accuracy is a little lower, the kappa of both decision tree and multinomial logistic regression are higher.

Accuracy: decision tree:O.195 multinomial logistic regression:O.176

kappa:decision tree:O.035 multinomial logistic regression:O.03

Table 3 - with cross-validation: Decision Tree

with cross-validation: decision tree

	Accuracy	Kappa
overall	0.195	0.035

	Sensitivity	Specificity	Balanced Accuracy
Class: Friday	0.000	1.000	0.500
Class: Monday	0.000	1.000	0.500
Class: Saturday	0.199	0.822	0.510
Class: Sunday	0.545	0.576	0.560
Class: Thursday	0.427	0.638	0.532
Class: Tuesday	0.000	1.000	0.500
Class: Wednesday	0.000	1.000	0.500

Table 4 - Multinomial Logistic Regression

multinomial logistic regression

	Accuracy	Kappa
overall	0.176	0.03

	Sensitivity	Specificity	Balanced Accuracy
Class: Friday	0.070	0.939	0.505
Class: Monday	0.114	0.895	0.504
Class: Saturday	0.121	0.886	0.504
Class: Sunday	0.340	0.759	0.550
Class: Thursday	0.225	0.806	0.515
Class: Tuesday	0.087	0.907	0.497
Class: Wednesday	0.190	0.839	0.514

Weekdays and Weekend-Balanced data with Cross-Validation

According to the table5, the balanced accuracy and kappa is higher after we aggregate the days in a week into weekday and weekends. We will finally choose logistic regression model for it has highest balanced accuracy and kappa.

Table 5 - The scores of three models

The scores of three models (Decison Tree, Random Forest, Logistic Regression)
	Sensitivity	Specifity	Accuracy	Balanced_accuracy	Kappa
Decison Tree	0.488	0.614	0.533	0.551	0.091
Random Forest	0.609	0.489	0.567	0.549	0.093
Logistic Regression	0.628	0.479	0.575	0.554	0.103

Unsupervised learning

Clustering

Clustering is a method that divides categorized or uncategorized data into similar groups or clusters. We used 2 methods for clustering, including Hierarchical clustering and Partitioning, and chose the number of clusters from statistics. We found that the number of clusters highly depend on the statistics chosen. For example, for the Partitioning method, the optimal number of clusters from Gap statistic is 13, while the optimal number from Silhouette is only 2.

For Hierarchical clustering, we chose 3 as an optimal number of clusters. The main characteristics of each cluster are, for example, high “produce” for cluster 1, high “snacks” and “beverages” for cluster 3. The average numbers of purchases from cluster 1 and 3 are significantly higher than cluster 2 in every department.

For Partitioning methods, we chose 2 as an optimal number of clusters for both K-mean and PAM models. For K-mean, we found that cluster 2 has higher average numbers of purchases than cluster 1 in every department. This means that cluster 2 in K-mean has similar characteristics as cluster 1 and 3 from the Hierarchical clustering. For PAM, the results are very close to the results from K-mean.

PCA

To derive the benefits of PCA and to test our hypothesis that grouping aisles reflects purchasing customer behavior better than grouping departments, we perform PCA by aisles. Non-scaled PCA is our focus to set the environment in line with the supervised learning analysis. As for the result, component1 to component28 (out of 134 components) can explain 75.5% of the total variation of the data. Fresh vegetable, fresh fruits and packaged vegetables fruits are the top three most variance and contributors to component1 and 2. Scaled PCA is also performed and we need 93 components out of 134 components to explain at least 75% of the total variation. Thus, we cannot derive the benefits of PCA and it also shows that each aisle is not correlated. Afterward, hybrid supervised and unsupervised learning approach grabs our attention. The 1-28 components are applied to the best model from supervised learning approach which is the logistic regression model. However, we find that the hybrid method doesn’t improve the accuracy of the prediction.

Limitation

The data set provided by Instacart contains only a few variables, including products purchased per order, product description, day of the week of orders, and hours of orders. We found that this information is still not sufficient to create predictive models with high accuracy for commercial uses. However, Amazon is an ideal case study showing that, with high quality and comprehensive data set, it’s even possible to use machine learning for a predictive analysis to anticipate and ship products before customers actually order it.

We also find the limitation when running machine learning functions in R with too many observations. In our study, we focus only on the training data set, which contains over 100k observations and do not include the prior data set, which represents all historical orders before the training set and contains over a million observations. This is because the execution time is extremely long and there are errors regularly when we run the models with all historical data (prior + training).

Further study

Machine learning and Artificial intelligence have play a key role in the boom of the e-commerce industry. There are a many applications that we can apply machine learning techniques in this industry. For example, Purchase and Repurchase prediction for predicting what orders customers will purchase next, Recommendation system for recommending products that users might would like to purchase. This system is very important behind the success of Amazon and even Netflix. There are also other applications such as Fraud prediction and Marketing campaign. However, in order to build these models with sufficient accuracy for commercial use, it’s important to we need more data such as customer personal information, browsing history, click history to get a better understanding of customer behaviors.

References

Ivo Bernardo, Classification Decision Trees, Easily Explained, Aug 30, 2021

Saikumar Talari, Random Forest® vs Decision Tree: Key Differences, February 18, 2022

Wikipedia, Multinomial logistic regression, 23 August 2021

S. Walusala W, R. Rimiru, C. Otieno, A hybrid machine learning approach for credit scoring using PCA and logistic regression, International journal of computer, ISSN 2307-4523.

Jeremy Stanley, 3 Million Instacart Orders, Open Sourced, May 3, 2017

Bigcommerce Blog, Nick Shaw, Ecommerce Machine Learning: AI’s Role in the Future of Online Shopping

The Economic Time, Amazon may predict and ship your order before you place it, Jan 27, 2014

Instacart Market Basket Analysis

Ting Yang, Jiaming Huang, Manunpat Sirijaturaporn, and Arturo García Luna Beltrán

24 mayo, 2022

Introduction

Overview and Motivation

Research questions

Exploratory Data Analysis

Data Description

Table 1 - aisles

Table 2 - departments

Table 3 - products

Table 4 - order_products_train

Table 5 - purchase time per order table

Table 6 - user_purchases

Visualization

Distribution of the order by time of purchase

Top 10 number of purchase by aisle

The number of purchase by department

Sales Patterns

PCA by department

Non-scaled PCA (Covariance)

Scaled PCA (Correlation)

Supervised Learning

Data Preparation for Models

Models

Decision Trees - Classification

Random Forest

Multinomial logistic regression

Logistic regression

Variable Importance

Unsupervised learning

Clustering

Hierarchical clustering

Partitioning methods

PCA and hybrid supervised and unsupervised learning approach

PCA by aisle

A hybrid supervised and unsupervised learning approach

Conclusion

Supervised learning

Unsupervised learning

Limitation

Further study

References